use vstd::prelude::*;
fn main() {}
verus!{



proof fn f1_lemma_vec_push<T>(vec: Seq<T>, i: T, l: usize)
    requires
        l == vec.len(),
    ensures
        forall|k: int| 0 <= k < vec.len() ==> #[trigger] vec[k] == vec.push(i)[k],
        vec.push(i).index(l as int) == i,
{
}

fn f1_contains(arr: &Vec<i32>, key: i32) -> (result: bool)
{
    let mut index = 0;
    while index < arr.len()
    {
        if (arr[index] == key) {
            return true;
        }
        index += 1;
    }
    false
}

fn f1_find_dissimilar(arr1: &Vec<i32>, arr2: &Vec<i32>) -> (result: Vec<i32>)
{
    let mut result = Vec::new();

    let mut index = 0;
    while index < arr1.len()
    {
        if (!f1_contains(arr2, arr1[index]) && !f1_contains(&result, arr1[index])) {
            
            result.push(arr1[index]);

        }
        index += 1;
    }
    let mut index = 0;
    while index < arr2.len()
    {
        if (!f1_contains(arr1, arr2[index]) && !f1_contains(&result, arr2[index])) {
            
            result.push(arr2[index]);
        }
        index += 1;
    }

    result
}





fn f2_insert_before_each(arr: &Vec<i32>, elem: i32) -> (result: Vec<i32>)
{
    let mut result: Vec<i32> = Vec::new();
    let mut index = 0;

    while index < arr.len()
    {
        result.push(elem);
        result.push(arr[index]);
        index += 1;
    }
    result
}





fn f3_has_only_one_distinct_element(arr: &Vec<i32>) -> (result: bool)
{
    if arr.len() <= 1 {
        return true;
    }
    let mut index = 1;
    while index < arr.len()
    {
        if arr[0] != arr[index] {
            return false;
        }
        index += 1;
    }
    true
}



spec fn step1(__hp0: Seq<i32>, __hp1: Seq<i32>, outp: Seq<i32>) -> bool {
    (forall|i: int|
            0 <= i < __hp0.len() ==> (!__hp1.contains(#[trigger] __hp0[i]) ==> outp.contains(
                __hp0[i],
            )))
    && (forall|i: int|
            0 <= i < __hp1.len() ==> (!__hp0.contains(#[trigger] __hp1[i]) ==> outp.contains(
                __hp1[i],
            )))
    && (forall|i: int, j: int|
            0 <= i < j < outp.len() ==> #[trigger] outp[i] != #[trigger] outp[j])
}

spec fn step2(inp: Seq<i32>, p2_elem: i32, outp: Seq<i32>) -> bool {
    (outp.len() == (2 * inp.len()))
    && (forall|k: int| 0 <= k < inp.len() ==> #[trigger] outp[2 * k] == p2_elem)
    && (forall|k: int| 0 <= k < inp.len() ==> #[trigger] outp[2 * k + 1] == inp[k])
}

spec fn step3(inp: Seq<i32>, outp: bool) -> bool {
    (outp == (forall|i: int| 1 <= i < inp.len() ==> inp[0] == #[trigger] inp[i]))
}

fn chain_driver(arr1: &Vec<i32>, arr2: &Vec<i32>, p2_elem: i32) -> (r_final: bool)
    ensures
        exists|v1: Seq<i32>, v2: Seq<i32>| #[trigger] step1(arr1@, arr2@, v1) && #[trigger] step2(v1, p2_elem, v2) && step3(v2, r_final),
{
    let x1 = f1_find_dissimilar(arr1, arr2);
    let x2 = f2_insert_before_each(&x1, p2_elem);
    let x3 = f3_has_only_one_distinct_element(&x2);
    proof {
        assert(step1(arr1@, arr2@, x1@));
        assert(step2(x1@, p2_elem, x2@));
        assert(step3(x2@, x3));
    }
    x3
}

} // verus!